Skip to main content
SpringerLink
Log in

Averaging in Cascade Junctions with a “Wide” Transmission Domain

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

Abstract

We consider a boundary-value problem for the Poisson equation in a dense multilevel junction consisting of a body of the junction and many periodically situated thin rectangles connected with the body by a transmission layer with a periodic boundary. It is supposed that the rectangles have finite lengths and the transmission layer has a small width that is much greater than the period. Nonhomogeneous Neumann boundary conditions are imposed upon the boundary of the transmission layer. An addition parameter of perturbation is included in these boundary conditions. Averaged problems are constructed and convergence theorems for solution and energy integrals are proved as a function of that parameter.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Y. Amirat, G. A. Chechkin, and R. R. Gadyl’shin, “Asymptotics of simple eigenvalues and eigenfunctions for the Laplace operator in a domain with oscillating boundary,” Comput. Math. Math. Phys., 46, 97–110 (2006).

    Article  MathSciNet  Google Scholar 

  2. Y. Amirat, G. A. Chechkin, and R. R. Gadyl’shin, “Asymptotics for eigenelements of Laplacian in domain with oscillating boundary: multiple eigenvalues,” Appl. Anal., 86, No. 7, 873–897 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  3. Y. Amirat, G. A. Chechkin, and R. R. Gadyl’shin, “Asymptotics of the solution of a Dirichlet spectral problem in a junction with highly oscillating boundary,” C. R. Méc., 336, No. 9, 693–698 (2008).

    Article  MATH  Google Scholar 

  4. Y. Amirat, G. A. Chechkin, and R. R. Gadyl’shin, “Asymptotic approximation of eigenelements of the Dirichlet problem for the Laplacian in a domain with shoots,” Math. Methods Appl. Sci., 33, 811–830 (2010).

    MathSciNet  MATH  Google Scholar 

  5. C. D’Apice, U. De Maio, and T. A. Mel’nik, “Asymptotic analysis of a perturbed parabolic problem in a thick junction of type 3:2:2,” Netw. Heterog. Media, 2, 255–277 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  6. N. S. Bakhvalov and G. P. Panasenko, Averaging Processes in Periodic Media. Mathematical Problems of the Mechanics of Composite Materials [in Russian], Nauka, Moscow (1984).

    Google Scholar 

  7. A. G. Belyaev, On Singular Perturbations of Boundary-value Problems, Doctoral thesis, Moscow, 1990.

  8. A. G. Belyaev, A. L. Pyatnitskii, and G. A. Chechkin, “Asymptotic behavior of a solution of a boundary-value problem in a perforated domain with oscillating bondary,” Sib. Math. J., 39, No. 4, 621–644 (1998).

    Article  MathSciNet  Google Scholar 

  9. A. Bensoussan, J.-L. Lions, and G. Papanicolau, Asymptotic Analysis for Periodic Structures, North-Holland Publishing Co., Amsterdam (1978).

    MATH  Google Scholar 

  10. D. Blanchard, A. Gaudiello, and G, Griso, “Junction of a periodic family of elastic rods with 3D plate. Part I,” J. Math. Pures Appl., (9) 88, No. 9, 1–33 (2007).

    MathSciNet  MATH  Google Scholar 

  11. D. Blanchard, A. Gaudiello, and G. Griso, “Junction of a periodic family of elastic rods with 3D plate. Part II,” J. Math. Pures Appl. (9), 88, No. 9, 149–190 (2007).

    MathSciNet  MATH  Google Scholar 

  12. D. Blanchard, A. Gaudiello, and T. A. Mel’nyk, “Boundary homogenization and restriction of dimension in a Kirchhoff-Love plate,” SIAM J. Math. Anal., 39, No. 6, 1764–1787 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  13. D. Blanchard, A. Gaudiello, and J. Mossino, “Highly oscillating boundaries and restriction of dimension: the critical case,” Anal. Appl. (Singap.), 5, 137–163 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  14. G. Bouchitté, A. Lidouh, and P Suquet, “Homogénéisation de frontière pour la modélisation du contact entre un corps déformable non linéaire et un corps rigide,” C. R. Acad. Sci. Paris. Sér. I, 313, 967–972 (1991).

  15. G. A. Chechkin and T. P. Chechkina, “On the homogenization of problems in domains of “Infusoria” type,” J. Math. Sci., 120, 1470–1482 (2004).

    Article  MathSciNet  Google Scholar 

  16. G. A. Chechkin and T. P. Chechkina, “An averaging theorem for problems in domains of “infusoria” type with inconsistent structure,” J. Math. Sci., 123, 4363–4380 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  17. G. A. Chechkin, T. P. Chechkina, C. D’Apice, U. De Maio, and T. A. Mel’nik, “Asymptotic analysis of a boundary value problem in a cascade thick junction with a random transmission Zone,” Appl. Anal., 88, Nos. 10–11, 1543–1562 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  18. G. A. Chechkin, T. P. Chechkina, C. D’Apice, U. De Maio, and T. A. Mel’nik, “Homogenization of 3D thick cascade junction with the random transmission zone periodic in one direction,” Russ. J. Math. Phys., 17, No. 1, 35–55 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  19. G. A. Chechkin and D. Cioranescu, “Vibration of a thin plate with a “rough” surface,” Stud. Math. Appl., 31, 147–169 (2002).

    Article  MathSciNet  Google Scholar 

  20. G. A Chechkin, A. Friedman, and A. L. Piatnitski, “The boundary-value problem in domains with very rapidly oscillating boundary,” J. Math. Anal. Appl., 231, No. 1, 213–234 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  21. G. A. Chechkin and T. A. Mel’nik, “Homogenization of a boundary value problem in a thick cascade junction,” J. Math. Sci., 154, 50–77 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  22. G. A. Chechkin and T. A. Mel’nik, “Asymptotic analysis of boundary value problems in thick cascade junctions,” Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki, 9, 16–22 (2008).

    MathSciNet  Google Scholar 

  23. G. A. Chechkin and T. A. Mel’nik, “Asymptotic analysis of boundary value problems in thick three-dimensional multi-level junctions,” Sb. Math., 200, Nos. 3–4, 357–383 (2009).

    MathSciNet  MATH  Google Scholar 

  24. G. A. Chechkin, A. L. Pyatnitskiy, and A. S. Shamaev, Averaging Methods and Applications, Novosibirsk (2007).

  25. T. P. Chechkina, G. A. Chechkin, and T. A. Mel’nik, “Convergence theorem for solutions and energy integrals of boundary value problems in thick multi-level junctions of a new type with perturbed Neumann conditions on the boundary of thin rectangles,” J. Math. Sci., 159, 113–132 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  26. D. Cioranescu and J. Saint Jean Paulin, Homogenization of Reticulated Structures, Springer- Verlag New York (1999).

    Book  MATH  Google Scholar 

  27. E. N. Dancer and D. Daners, “Domain perturbation for elliptic equations subject to Robin boundary conditions,” J. Differ. Equ., 138, No. 1, 86–132 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  28. U. De Maio, T. Durante, and T. A. Mel’nik, “Asymptotic approximation for the solution to the robin problem in a thick multi-level junction,” Math. Models Methods Appl. Sci., 15, No. 12, 1897–1921 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  29. T. Durante T and T. A. Mel’nik, “Asymptotic analysis of a parabolic problem in a thick two-level junction,” Zh. Mat. Fiz. Anal. Geom., 3, No. 3, 313–341 (2007).

    MathSciNet  MATH  Google Scholar 

  30. T. Durante, T. A. Mel’nik, and P. S. Vashchuk, “Asymptotic aproximation for the solution to a boundary-value problem with varying type of boundary conditions in a thick two-level junction,” Nonlinear Oscil., 9, No. 3, 336–355 (2006).

    Article  MathSciNet  Google Scholar 

  31. A. Gaudiello, “Asymptotic behavior of non-homogeneous Neumann problems in domains with oscillating boundary,” Ric. Mat., 43, 239–292 (1994).

    MathSciNet  MATH  Google Scholar 

  32. W. JägerWand A.Mikelić, “On the roughness-induced boundary conditions for an incompressible viscous flow,” J. Differ. Equ., 170, No. 1, 96–122 (2001).

    Article  Google Scholar 

  33. Yu. A. Kazmerchuk and T. A. Mel’nyk, “Homogenization of the Signorini boundary-value problem in a thick plane junction,” Nonlinear Oscil., 12, No. 1, 45–59 (2009).

    Article  MathSciNet  Google Scholar 

  34. J. B. Keller and J. Nevard, “Homogenization of rough boundaries and interfaces,” SIAM J. Appl. Math., 57, No. 6, 1660–1686 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  35. E. Ya. Khruslov, “On resonance conditions in a diffraction problem,” Teor. Funkts. Funkts. Anal. Prilozh., 6, 111–129 (1968).

    MATH  Google Scholar 

  36. E. Ya. Khruslov and V. P. Kotlyakov, “On a limit boundary conditions of a Neumann problem,” Teor. Funkts. Funkts. Anal. Prilozh., 10, 83–96 (1970).

  37. E. Ya. Khruslov and V. A. Marchenko, Boundary-Value Problems in Domains with a Small- Grained Boundary, Naukova Dumka, Kiev (1974).

    Google Scholar 

  38. W. Kohler, G. Papanicolaou, and S. Varadhan, “Boundary and interface problems in regions with very rough boundaries,” Multiple Scattering and Waves in Random Media, North-Holland, Amsterdam (1981).

  39. O. A. Ladyzhenskaya (ed.), Boundary-Value Problems of Mathematical Physics, 14, Nauka, Moscow (1990).

    Google Scholar 

  40. T. A. Mel’nik, “Averaging of elliptic equations that describe processes in strongly inhomogenoeus thin punctured domains with rapidly changing thickness,” Dokl. Akad. Nauk Ukr. SSR, 10, 15–18 (1991).

    MathSciNet  Google Scholar 

  41. T. A. Mel’nik, “Homogenization of the Poisson equation in a thick periodic junction,” Z. Anal. Anwendungen, 18, No. 4, 953–975 (1999).

    MathSciNet  Google Scholar 

  42. T. A. Mel’nik, “Homogenization of a singularly perturbed parabolic problem in a thick periodic junction of type 3:2:1,” Ukr. Math. J., 52, No. 11, 1737–1749 (2000).

    Article  MathSciNet  Google Scholar 

  43. T. A. Mel’nik, “Asymptotic behavior of eigenvalues and eigenfunctions of the Fourier problem in a thick multi-level junction,” Ukr. Math. J., 58, No. 2, 220–243 (2006).

    Article  MathSciNet  Google Scholar 

  44. T. A. Mel’nik, “Homogenization of a boundary-value problem with a nonlinear boundary condition in a thick junction of type 3:2:1,” Math. Models Methods Appl. Sci., 31, No. 9, 1005–1027 (2008).

    Article  MathSciNet  Google Scholar 

  45. T. A. Mel’nik and S. A. Nazarov, “Asymptotic strusture of the spectrum in a problem of harmonic vibrations of a hub with heavy spokes,” Russ. Acad. Sci. Dokl. Math., 48, No. 3, 428–432 (1994).

    MathSciNet  Google Scholar 

  46. T. A. Mel’nik and S. A. Nazarov, “Asymptotic behavior of the solution of the Neumann spectral problem in a domain of “tooth comb” type,” J. Math. Sci., 85, 2326–2346 (1997).

    Article  MathSciNet  Google Scholar 

  47. T. A. Mel’nik and S. A. Nazarov, “Asymptotic analysis of the Neumann problem on the junction of a body and thin heavy rods,” St. Petersburg Math. J., 12, No. 2, 317–351 (2001).

    MathSciNet  Google Scholar 

  48. T. A. Mel’nik and P. S. Vaschuk, “Homogenization of a boudnary-value problem with boundary conditions of mixed type in a thick junction,” Differ. Equ., 43, No. 5, 696–703 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  49. S. A. Nazarov, “Junctions of singularly degenerating domains with different limit dimensions. Part I,” J. Math. Sci., 80, 1989–2034 (1999).

    Article  Google Scholar 

  50. S. A. Nazarov, “Junctions of singularly degenerating domains with different limit dimensions. Part II,” J. Math. Sci., 97, 155–195 (1999).

    Google Scholar 

  51. S. A. Nazarov, Asymptotic Theory of Thin Plates and Rectangles [in Russian], Nauchnaya Kniga, Novosibirsk (2002).

    Google Scholar 

  52. S. L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics [in Russian], Nauka, Moscow (1988).

    Google Scholar 

  53. S. L. Sobolev, Selected Questions of Theory of Functional Spaces and Generalized Functions [in Russian], Nauka, Moscow (1989).

    Google Scholar 

  54. V. V. Zhikov, S. M. Kozlov, and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin (1994).

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. National Research Nuclear Institute “MIFI”, Moscow, Russia

    T. P. Chechkina

Authors
  1. T. P. Chechkina
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to T. P. Chechkina.

Additional information

Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 39, Partial Differential Equations, 2011.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chechkina, T.P. Averaging in Cascade Junctions with a “Wide” Transmission Domain. J Math Sci 190, 157–169 (2013). https://doi.org/10.1007/s10958-013-1251-7

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-013-1251-7

Keywords

Search

Navigation